Input: orange shaft. Lead of its helical slot is Li.
Output: green shaft (coaxial with the input). Lead of its helical slot is Lo.
Output lead angle should not be less than 30 deg. to ease the output motion (to avoid self-locking).
Pink slider has a pin that contacts slots of both shafts.
Transmission ratio: i = Lo/Li
The two shafts turn in the same direction if their helical slots are of the same hand and vice versa.
The video shows case when Lo = 2 Li. Thread hands are opposite.
The two shafts turn in opposite directions. The orange turns 1 rev., the green turns ½ rev.
If Lo = Li, the mechanism becomes a coaxial reverser.